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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory
Subject: Re: Real Number --- Merely numbers whose digits can be infinitely long
Date: Wed, 01 May 2024 20:46:00 -0700
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Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
> wij <wyniijj5@gmail.com> writes:
>> On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
>>> wij <wyniijj5@gmail.com> writes:
>>> > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
>>> > > wij <wyniijj5@gmail.com> writes:
>>> [...]
>>> > > >      <fixed_point_number>::= [-] <wnum> [ . <frac> ]  // excluding "-0" case
>>> > > >      <wnum>::= 0
>>> > > >      <wnum>::= <nzd> { 0 | <nzd> }
>>> > > >      <frac>::= { 0 | <nzd> } <nzd>
>>> > > >      <nzd> ::= 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 // 'digit' varys depending on n-ary
>>> > > > 
>>> > > >     Ex: 78, -12.345, 3.1414159
>>> > > 
>>> > > So what's the point of defining these strings that represent a subset of
>>> > > the rationals?
>>> > 
>>> > <fixed_point_number> is a super set of rationals.
>>> [...]
>>> 
>>> An extraordinary claim.
>>> 
>>> Do you agree that 1/3 is a rational number?  How is 1/3 represented in
>>> your <fixed_point_number> notation?
>>> 
>>
>> I already told you: 1/3= 0.1 (3-ary <fixed_point_number>)
>> Substitute the n in n-ary with the q in p/q, every p/q is representable 
>> by <fixed_point_number>.
>> And, the rule of <frac> can generate infinitely long fractions, read it carefully!
>
> That kind of notation almost universally refers to *finite* sequences of
> symbols.
>
> If you intend it to be able to specify infinite sequences, that's fine,
> but it's not inherent in the notation you've presented.  I also wonder
> how an infinitely long <frac> can have <nzx> as its last element.
>
> So <frac> can be infinitely long.  Can <wnum> be infinitely long?
>
> I presume that the "n-ary" base can be any integer greater than or equal
> to 2, and that the digits can range from 0 to n-1.  That means you'll
> need arbitrarily many distinct symbols for the digits in large bases.
> That's all fine, but it would be good to state all this explicitly.
>
> There are already perfectly good mathematical methods for constructing
> the integers, the rationals, and the reals.  Your method of using base-n
> notation to *define* the reals and/or rationals seems superfluous.  It
> can probably be done consistently, but I fail to see how it's useful.

And something I thought of immediately after I posted the above:

You need to use different bases to represent all rational numbers, but
the base isn't part of your notation.  Your grammar matches "0.1", but
how do I know whether than's 1/10, 1/3, or 1/1729?

0.2 (base 10) and 0.1 (base 5) represent the same number.  0.2 (base 10)
and 0.1 (base 4) do not.  Your notation doesn't seem to have any way to
indicate this.  How can we know that 0.2 (base 10) and 0.1 (base 5) are
equal without using the real numbers that you're trying to *define*?

Or are you assuming that real numbers already exist, and you're defining
this notation on top of that?  If so, what's the point?

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
void Void(void) { Void(); } /* The recursive call of the void */